SPANNING SIMPLICIAL COMPLEXES OF UNI-CYCLIC MULTIGRAPHS IMRAN AHMED, SHAHID MUHMOOD Abstract. A multigraph is a nonsimple graph which is permitted to have multiple edges, that is, edges that have the same end nodes. We introduce the concept of spanning simplicial complexes ∆s(G) of multigraphs G, which provides a generalization of spanning simplicial complexes of associated simple graphs. We give first the characterization of all spanning trees of a r n r uni-cyclic multigraph Un,m with edges including multiple edges within and outside the cycle of length m. Then, we determine the facet ideal I r r F (∆s(Un,m)) of spanning simplicial complex ∆s(Un,m) and its primary decomposition. The Euler characteristic is a well-known topological and homotopic invariant to classify surfaces. Finally, we device a formula for r Euler characteristic of spanning simplicial complex ∆s(Un,m). Key words: multigraph, spanning simplicial complex, Euler characteristic. 2010 Mathematics Subject Classification: Primary 05E25, 55U10, 13P10, Secondary 06A11, 13H10. 1. Introduction Let G = G(V, E) be a multigraph on the vertex set V and edge-set E. A spanning tree of a multigraph G is a subtree of G that contains every vertex of G. We represent the collection of all edge-sets of the spanning trees of a multigraph G by s(G). The facets of spanning simplicial complex ∆s(G) is arXiv:1708.05845v1 [math.AT] 19 Aug 2017 exactly the edge set s(G) of all possible spanning trees of a multigraph G. Therefore, the spanning simplicial complex ∆s(G) of a multigraph G is defined by ∆s(G)= hFk | Fk ∈ s(G)i, which gives a generalization of the spanning simplicial complex ∆s(G) of an associated simple graph G. The spanning simplicial complex of a simple con- nected finite graph was firstly introduced by Anwar, Raza and Kashif in [1]. Many authors discussed algebraic and combinatorial properties of spanning simplicial complexes of various classes of simple connected finite graphs, see for instance [1], [5], [6] and [9]. Let ∆ be simplicial complex of dimension d. We denote fi by the number of i-cells of simplicial complex ∆. Then, the Euler characteristic of ∆ is given 1 2 Ahmed and Muhmood by d−1 i χ(∆) = (−1) fi, Xi=0 which is a well-known topological and homotopic invariant to classify surfaces, see [4] and [7]. r The uni-cyclic multigraph Un,m is a connected graph having n edges includ- ing r multiple edges within and outside the cycle of length m. Our aim is to give some algebraic and topological characterizations of spanning simplicial r complex ∆s(Un,m). In Lemma 3.1, we give characterization of all spanning trees of a uni-cyclic r multigraph Un,m having n edges including r multiple edges and a cycle of r length m. In Proposition 3.2, we determine the facet ideal IF (∆s(Un,m)) of r spanning simplicial complex ∆s(Un,m) and its primary decomposition. In The- orem 3.3, we give a formula for Euler characteristic of spanning simplicial r complex ∆s(Un,m). 2. Basic Setup A simplicial complex ∆ on [n] = {1,...,n} is a collection of subsets of [n] satisfying the following properties. (1) {j} ∈ ∆ for all j ∈ [n]; (2) If F ∈ ∆ then every subset of F will belong to ∆ (including empty set). The elements of ∆ are called faces of ∆ and the dimension of any face F ∈ ∆ is defined as |F |−1 and is written as dim F , where |F | is the number of vertices of F . The vertices and edges are 0 and 1 dimensional faces of ∆ (respectively), whereas, dim ∅ = −1. The maximal faces of ∆ under inclusion are said to be the facets of ∆. The dimension of ∆ is denoted by dim ∆ and is defined by dim ∆ = max{dim F | F ∈ ∆}. If {F1,...,Fq} is the set of all the facets of ∆, then ∆ =< F1,...,Fq >. A simplicial complex ∆ is said to be pure, if all its facets are of the same dimension. A subset M of [n] is said to be a vertex cover for ∆ if M has non-empty intersection with every Fk. M is said to be a minimal vertex cover for ∆ if no proper subset of M is a vertex cover for ∆. Definition 2.1. Let G = G(V, E) be a multigraph on the vertex set V and edge-set E. A spanning tree of a multigraph G is a subtree of G that contains every vertex of G. Definition 2.2. Let G = G(V, E) be a multigraph on the vertex set V and edge- set E. Let s(G) be the edge-set of all possible spanning trees of G. We define a simplicial complex ∆s(G) on E such that the facets of ∆s(G) are exactly the elements of s(G), we call ∆s(G) as the spanning simplicial complex of G and given by ∆s(G)= hFk | Fk ∈ s(G)i. Spanning Simplicial Complexes of Uni-Cyclic Multigraphs 3 r Definition 2.3. A uni-cyclic multigraph Un,m is a connected graph having n edges including r multiple edges within and outside the cycle of length m. Let ∆ be simplicial complex of dimension d. Then, the chain complex C∗(∆) is given by − ∂d ∂d 1 ∂2 ∂1 ∂ 0 0 → Cd(∆)−→Cd−1(∆)−−−→...−→C1(∆)−→C0(∆)−→ 0. Each Ci(∆) is a free abelian group of rank fi. The boundary homomorphism ∂d : Cd(∆) → Cd−1(∆) is defined by d d i d ∂d(σα)= i=0(−1) σα|[v0,v1,...,vi,...,vˆ d]. Of course, P Hi(∆) = Zi(∆)/Bi(∆) = Ker∂i/Im∂i+1, where Zi(∆) = Ker ∂i and Bi(∆) = Im ∂i+1 are the groups of simplicial i- cycles and simplicial i-boundaries, respectively. Therefore, rank Hi(∆) = rank Zi(∆) − rank Bi(∆). One can easily see that rank Bd(∆) = 0 due to Bd(∆) = 0. For each i > 0 there is an exact sequence ∂i 0 → Zi(∆) → Ci(∆)−→Bi−1(∆) → 0. Moreover, fi = rank Ci(∆) = rank Zi(∆) + rank Bi−1(∆). Therefore, the Euler characteristic of ∆ can be expressed as d i d i χ(∆) = i=0(−1) fi = i=0(−1) (rank Zi(∆) + rank Bi−1(∆)) d i d i = i=0(P−1) rank Zi(∆)P + i=0(−1) rank Bi−1(∆). ChangingP index of summationP in the last sum and using the fact that rank B−1(∆)=0= rank Bd(∆), we get d i d i+1 χ(∆) = i=0(−1) rank Zi(∆) + i=0(−1) rank Bi(∆) d i d i = i=0(P−1) (rank Zi(∆) − rankP Bi(∆)) = i=0(−1) rank Hi(∆). Thus,P the Euler characteristic of ∆ is givenP by d i χ(∆) = (−1) βi(∆), Xi=0 where βi(∆) = rank Hi(∆) is the i-th Betti number of ∆, see [4] and [7]. r 3. Topological Characterizations of ∆s(Un,m) r Let Un,m be a uni-cyclic multigraph having n edges including r multiple edges within and outside the cycle of length m. We fix the labeling of the edge r set E of Un,m as follows: ′ ′ ′ E = {e11,...,e1t1 ,...,er 1,...,er tr′ , e(r +1)1,...,em1, e(m+1)1,...,e(m+1)tm+1 , ′′ ′′ ′′ ...,e(m+r )1,...,e(m+r )tm+r , e1,...,ev}, where ei1,...,eiti are the multiple ′ edges of i-th edge of cycle with 1 ≤ i ≤ r while e(r′+1)1,...,em1 are the single edges of the cycle and ej1,...,ejtj are the multiple edges of the j-th ′′ edge outside the cycle with m +1 ≤ j ≤ m + r , moreover, e1,...,ev are single edges appeared outside the cycle. 4 Ahmed and Muhmood r We give first the characterization of s(Un,m). r Lemma 3.1. Let Un,m be the uni-cyclic multigraph having n edges including r multiple edges and a cycle of length m with the edge set E, given above. A r ′ subset E(Twiw ) ⊂ E will belong to s(Un,m) if and only if Twiw = {e1i1 ,...,er ir′ , ′ ′′ ′′ for some e(r +1)1,...,em1, e(m+1)i(m+1) ,...,e(m+r )i(m+r ) , e1,...,ev}\{ewiw } ih ∈ ′ ′′ {1,...,th} with 1 ≤ h ≤ r , m +1 ≤ h ≤ m + r and iw ∈ {1,...,tw} with ′ ′ 1 ≤ w ≤ r or iw = 1 with r +1 ≤ w ≤ m for some w = h and iw = ih appeared in Twiw . r Proof. By cutting down method [3], the spanning trees of Un,m can be obtained ′ by removing exactly th −1 edges from each multiple edge such that 1 ≤ h ≤ r , m +1 ≤ h ≤ m + r′′ and in addition, an edge from the resulting cycle need to be removed. Therefore, the spanning trees will be of the form Twiw = ′ ′ ′ ′′ ′′ {e1i1 ,...,er ir , e(r +1)1,...,em1, e(m+1)i(m+1) ,...,e(m+r )i(m+r ) , e1,...,ev}\{ewiw } ′ ′′ for some ih ∈ {1,...,th} with 1 ≤ h ≤ r , m + 1 ≤ h ≤ m + r and ′ ′ iw ∈ {1,...,tw} with 1 ≤ w ≤ r or iw = 1 with r + 1 ≤ w ≤ m for some w = h and iw = ih appeared in Twiw . In the following result, we give the primary decomposition of facet ideal r IF (∆s(Un,m). r Proposition 3.2. Let ∆s(Un,m) be the spanning simplicial complex of uni- r cyclic multigraph Un,m having n edges including r multiple edges within and outside the cycle of length m. Then, r IF (∆s(Un,m)= (x ) (x ,...,x , x ) (x , x ) a i1 iti k1 k1 l1 1≤\a≤v \ 1≤i≤r′ ;\r′+1≤k≤m \ r′+1≤\k<l≤m (x ,...,x , x ,...,x ) (x ,...,x ) , i1 iti b1 btb j1 jtj \ 1≤i<b\≤r′ \ m+1≤\j≤m+r′′ ′ where ti with 1 ≤ i ≤ r is the number of multiple edges appeared in the i-th ′′ edge of the cycle and tj with m +1 ≤ j ≤ m + r is the number of multiple edges appeared in the jth-edge outside the cycle.